Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3,082 questions
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Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
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Quotients of the Hilbert space
Let $G$ be a compact Lie group with a biinvariant metric.
Note that $G\times G$ acts isometrically on $G$ from left and right.
Consider the quotient $D=G/H$ by a closed subgroup $H\le G\times G$;
if $...
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Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
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Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...
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Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
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Lower bound for the normal injectivity radius
Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a ...
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Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
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Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
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Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
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Collapse of Moebius bands with bounded below Gauss curvature and convex boundary
Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the ...
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The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
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Regularity of Metric when defining C^k norms
Given $(M^n, g)$ closed riemannian manifold, I am wondering about the definition of the $C^k$ norms with respect to the metric, and how these norms depend on $g$. For example, I would assume that if $...
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Second variation formula in Spivak, Volume 4
Let $\alpha: (-\varepsilon,\varepsilon) \times [0,1] \to M$ be a smooth variation of geodesics on a Riemannian manifold $M$, not necessarily fixed at endpoints. Then in Spivak, Volume 4, Chapter 8, ...
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Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
CommunityBot
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How badly does the geodesic exponential map fail to be $C^2$ on Finsler manifolds
Consider a Finsler manifold $M$. Then for each $x \in M$, we can consider the partial map $\exp_x: T_x M \to M$, which is $C^\infty$ away from the origin, $C^1$ at the origin, but never $C^2$ at the ...
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Motivation and examples of parabolic manifolds
Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...
CommunityBot
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Invariance of the Lebesgue measure
It is well known that the Lebesgue measure is the unique (up to a multiplicative constant) sigma-finite Borel measure on $\mathbb{R}^d$ which is translation invariant.
I am wondering if a similar ...
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Tensor component calculation
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please transfer to the relevant site.
Recall that in terms of Weyl and ...
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Property of parallel translation in Green and Wu, "On the subharmonicity and plurisubharmonicity of geodesically convex functions"
In the mentionned paper, I am having difficulties in understanding the proof of lemma 2.
Roughly, this lemma says that given any separation $\eta$ for the $C^\infty$ topology of smooth paths from $[-1,...
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Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
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Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
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Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a ...
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Is a Riemannian submersion a harmonic map?
Is every Riemannian submersion necessarily a Harmonic map? If not under what condition that is true?
The motivation: the linear part of a Riemannian submersion is the direct sum og an isometry ...
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Does isometric immersion map boundary to boundary?
Let $M$ be a compact, connected, oriented, smooth Riemannian manifold with non-empty boundary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion.
Is it true that $f(\partial M) \...
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Closed geodesic on Riemannian manifold
My question is ( an exercise of Riemannian geometry ):
Let $M$ be a complete Riemannian manifold, suppose there exists $c>0$ such that the sectional curvature of $M$ is greater than or equal to $c$...
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Curvature of geodesic spheres
Suppose $\mathbb{R}^d$ is given a Riemannian metric $g$. Fix a point $p \in \mathbb{R}^d$, fix two tangent vectors $v,w \in T_p \mathbb{R}^d$ with $\langle v, w \rangle_g = 0$, and consider the ...
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Injective hulls of metric spaces
In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
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Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
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Question on harmonic map between Riemannian manifolds
Let $S=[0,1]^2$. Ignoring issues having to do with boundaries and corners, a chart is a diffeomorphism $\varphi \colon S \to S$. Let $g_0$ denote the flat (euclidean) metric. Given a chart $\varphi \...
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Trace-free Hermitian endomorphisms in dimension $7$
Let $M$ be a spin-manifold of dimension $7$. Let $S\rightarrow M$ be a spin-bundle on $M$. Then Clifford multiplication ($c$) gives us the following isomorphism:
\begin{align*}
c:i\Lambda^2\oplus\...
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Examples of rigid open surfaces
In the celebrated book of Hilbert and Cohn-Vossen, the following sentence appears (p. 230):
Bending is impossible in the case of all closed convex surfaces, such as, for example, the ellipsoids. It is ...
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Isometry group of a left-invariant Riemannian metric on $\mathrm{SU}(2)$
Recall that
\begin{equation}
\mathbb{S}^3=\operatorname{SU}(2)=\left\{
\begin{pmatrix}
z&w\\
-\bar{w}&\bar{z}
\end{pmatrix}
,|z|^2+|w|^2=1
\right\}
\end{...
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Horizontal knots on 3 sphere
Motivation: First I present my motivation for this question but this motivational part is not my main question.
I participated in a talk on knot theory. Then I presented the following ...
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Esimate of the Levi form of distance squared function
Let $(X,\omega)$ be a compact Kähler manifold. Let $\widetilde{X}$ be the universal cover of $X$. We denote by $\omega$ abusively the pullback metric of $\omega$ on $\widetilde{X}$. Fix a base point ...
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Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
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Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
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Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
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CommunityBot
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Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
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Are all linear vector fields geodesible vector fields?
I had already asked this question in MSE then I ask here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix.
Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\...
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Smooth isometric immersions of the a hemisphere in $\mathbb R^3$
Let $S$ be a disc endowed with a spherical metric (constant positive Gaussian curvature). What is known about its smooth isometric immersions in $\mathbb R^3$?
By [GS20], an immersion is uniquely ...
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The Frobenius integrability of distrbution and Hyers–Ulam–Rassias stability
Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is ...
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Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
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Are Bergman metrics on compact Riemann surfaces continuous on Teichmüller space?
Let $R$ be a compact Riemann surface of genus $\geq 1$, and let $\omega_1,\ldots,\omega_g$ be holomorphic one forms that form the dual basis of canonical homology basis. Let $(\pi)_{ij}$ be the ...
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Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
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When are these base spaces isomorphic?
Given a smooth manifold $\mathcal{M}$ that is a fibre bundle over two different base spaces, i.e., there are $\Pi_1:\mathcal{M}\rightarrow B_1$ and $\Pi_2:\mathcal{M}\rightarrow B_2$, if I can prove ...
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Upper bound on the sectional curvature of a Riemannian submersion
Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, endowed with the product metric given by the bi-invariant metric of $\operatorname{SO}(n)$ and the round metric of $\mathbb{S}...
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Lengths of closed geodesics and geodesic segments
Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:
existence of closed geodesics of arbitrarily long length on $M$...
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On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
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Mean-value type property for eigenfunctions of Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the ...
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Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$:
\begin{align*} A(t) &= \begin{bmatrix}e^t &...
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